Dynamical gauge fields

Engineering dynamical gauge fields in atomic and
optical systems: challenges and perspectives
GGI - What’s next - 05/05/2015
Marcello Dalmonte
Institute for Quantum Optics and
Quantum Information, and
Institute for Theoretical Physics,
Innsbruck
Joint work with A. Glätzle, P. Hauke, P. Zoller (Innsbruck), R. Nath (Innsbruck-Pune), R.
Moessner, I. Rousochatzakis (Dresden), I. Bloch, C. Gross (MPQ Garching) E. Rico
(Bilbao), S. Montangero, T. Pichler, P. Silvi (Ulm), P. Stebler, M. Boegli, U.-J. Wiese (Bern),
D. Banerjee (Desy), E. Martinez, D. Nigg, T. Monz, R. Blatt (Innsbruck), M. Mueller (Madrid)
AMO and HEP
This talk: why should
we search for connections
Quantum
between AMO
and
HEP
physics,
and
how
can
we
Chromodynamics
establish them?
Goal: qualitative overview to boost discussion with
toy models of lattice gauge theories:
experts from different
fields - what are the most
fields
✓ dynamical gauge
interesting perspectives
for this
field?
AMO
✓ … + coupled to matter fields
✓ Abelian and non-Abelian (U(N), SU(N))
✓real-time, statics,…
Outline
Why shall we try to establish connections between QI,
AMO and HEP?
1) observe new phenomena, explore new physics
2) use AMO systems as quantum simulators
Gauge theories in
HEP and
condensed matter
systems
Synthetic matter:
atoms, ions,
circuits, etc.
Key element: develop a framework to build this
connection. How?
Why? 1) Known and novel physics in the lab!
Experiments in condmat and HEP can be challenging - explore features
of gauge theories in controlled environments to check theories
Physics of toy models (‘t Hooft model, (walking) technicolor, many-spin
interactions in spin systems, ...) and correlated phenomenology (entanglement in
frustrated systems, string tension in confining theories, Higgs,....)
3
[V(r)-V(r0)] r0
2
β = 6.0
β = 6.2
β = 6.4
Cornell
String tension of
quenched QCD,
Bali’s review
PhysRep.2001
1
0
-1
-2
Entanglement
entropy of a Z-2
spin liquid,
Jiang et al.,
NatPhys.2012
-3
-4
0.5
1
1.5
r/r0
2
2.5
3
Cold atomic gases as controllable and tunable platforms
to observe and investigate novel many-body effects
Why? 2) From Classical to Quantum simulation of
gauge theories
Many phenomena in gauge theories stand beyond our
current capabilities
-> use synthetic systems as quantum simulators
Lattice QCD in (less than) a nutshell
quarks and gluons
a
lattice
non-perturbative approach to fundamental
theories of matter (e.g. QCD)
→ classical statistical mechanics
Montecarlo simulations are possible!
K. Wilson (1974)
Lattice QCD in (less than) a nutshell
Remarkable achievements
η
1)first evidence of quark-gluon plasma
2)ab-initio estimate of protonic mass
3)low-lying hadron spectrum
4)determination of CKM matrix
many more…
γ
1
a
quarks and gluons
lattice
non-perturbative approach to fundamental
theories of matter (e.g. QCD)
→ classical statistical mechanics
K. Wilson (1974)
∆md
∆ms
Summer14
SM fit
∆md
β
0.5
εK
0
Vub
Vcb
α
-0.5
BR(B→τ ν )
sin(2β+γ )
-1
-1
-0.5
0
0.5
1
ρ
UFit, utfit.org
BMW, Science 2015
A. Ukawa, Kenneth Wilson and lattice QCD, arXiv.1501.04215
BMW, Science 2008
Still, a lot of interesting open problems
Real-time dynamics: no
known reliable algorithm
Finite- and large-density
regimes: Montecarlo suffers
from severe sign problem
Ideal test-bed applications for quantum
simulators
Experiments in heavy ion
collider (LHC, Brookhaven, …)
- intrinsic real-time
dynamics, thermalization,
prethermalization, etc…
PSR B1509-58
Relevant for ab initio
nuclear physics,
astrophysics / color
superconductivity, …
A. Ukawa, Kenneth Wilson and lattice QCD, arXiv.1501.04215; U.-J. Wiese, Annalen der Physik 525, 777 (2013).
What is quantum simulation
Feynman’s lecture 'Simulating Physics with Computers’
0.6
0.6
0.4
0.4
nodd
nodd
“Nature isn't classical, dammit, and if you want to make a simulation of
NATURE PHYSICS
10.1038/NPHYS2232
nature,
you'dDOI:
better
make it quantum mechanical, andARTICLES
by golly it's a
wonderful problem, because
it doesn't look so easy.”
a
b
= 2.44(2)
Physics at 0.2
equilibrium U/J
(sign
K/J = 1 × 10
Motivation"
problem, entanglement,...)
0
0
1
2
3
4
5
U/J = 3.60(4)
K/J = 1.3 × 10¬2
1
2
3
4
5
4Jt/h
d
DMRG (theory)"
Breaks down here"
0.6
0.6
0.4
0.4
nodd
nodd
quench"
S. Trotzky et al., Nature
0.2Physics, 2232 (2012)"
U/J = 5.16(7)
K/J = 1.7 × 10¬2
0.2
0
0
● 
0
4Jt/h computation breaks down"
Quantum simulation where classical
c
"
Time-evolution of manyparticle quantum systems
0.2
¬2
● 
R. P. Feynman, Int. J. Theor. Phys. (1982).
1
2
3
4Jt/h
4
5
0
0
1
2
U/J = 9.9(1)
K/J = 2.9 × 10¬2
3
4
5
4Jt/h
Figure 2 | Relaxation of the local density for different interaction strengths. We plot the measured traces of the odd-site population n
(t) for four
odd
Develop different
toolsinteraction
to measure
in ensemble-averaged
AMO systems
(ion
chains?)"
strengths U/Jentanglement
(circles). The solid lines are
results from
t-DMRG
simulations without free parameters. The dashed
lines represent simulations including next-nearest neighbour hopping with a coupling matrix element JNNN /J ' 0.12 (a), 0.08 (b), 0.05 (c) and 0.03 (d)
Classical digital and analog computation
Analog classical simulators
Digital classical simulators
Antikythera
FermiAC
Pascaline
Wind tunnel
Supercomputers
Quantum digital and analog computation
Analog quantum simulators
Digital quantum simulators
Basic idea: Emulation - Build a Hamiltonian /
Lagrangian / Liouvillian using the following:
Basic idea: Trotterize a time-evolution
U (t)
e
iHt/
=e
iH
Synthetic systems
Spins
Bosons
Fermions
|0
b
†
...
iH
t1 /
S. Lloyd, Science (1996)
|1
↵
tn /
†
Wind tunnel
Platforms: cold atoms, ions, Rydbergs,
molecules, superconductors, …
Cirac and Zoller, NatPhys. 2012; Georgescu et al., RMP2014
Platforms: ions,
superconductors,
photonics, …
Ca40 quantum computer,
Rainer Blatt’s group
Supercomputers
Quantum
Annealers
Complementing classical and quantum simulations
η
Quantum simulators complement (and do not
substitute!!!!) classical simulations!
γ
1
∆md
∆ms
Summer14
SM fit
∆md
β
0.5
εK
0
Vub
Vcb
α
-0.5
BR(B→τ ν )
sin(2β+γ )
-1
-1
-0.5
0
0.5
1
ρ
Entanglement based numerical methods
use entanglement based
numerical
Moreover,
they methods to
simulate regimes which are
x
1) call for novel applications of known
techniques
(e.g. MonteCarlo)
for
challenging
for MonteCarlo,
e.g. 111,
dynamics
validation (see PRL
115303)
E. Rico, T. Pichler, MD, P.
2) are ideal ‘stimulators’ for novel classes
of algorithms
Zoller and
S. Montangero, - like t-dependent
t
PRL 112,
201601.
renormalization group (see works at Ulm,
MPQ,
ICFO, Gent, Innsbruck…)
However …
We need a generic
framework to establish
this connection: quantum
link models
High Energy - field theories
?
NB: some ad hoc solutions can
be found for specific models Wilson’s Schwinger model and
CP(N) models with finite t-angle
Synthetic systems
Spins
Bosons
Fermions
|1
|0
Review on dynamical gauge fields and quantum simulation: U.-J. Wiese, Annalen der Physik 525, 777 (2013).
Intermezzo - static vs dynamical gauge fields
Dynamical gauge fields: particles
hopping around a plaquette assisted
by additional link degrees of
freedom
Static gauge fields: particles
hopping around a plaquette acquire
a finite (Peierls) phase
4
3
¡=
˛
Ad s
L. Fallani M. Inguscio
Fermions
coupled
LENS, Florence
2 to
1
static gauge fields
x
x +Theory:
1
realized!
M. Rider,
,
arXiv.1502.02495
P. Zoller, MD
i 'x,x+1
(phase)
e
H = °t √†x e i 'x,x+1 √x+1 + h.c.
Theory Review: J. Dalibard et al., Rev. Mod. Phys. (2011)
Exp.: Munich, Hamburg, NIST/Maryland, Florence, ETH, …
Theory: many active groups
Dynamical gauge fields and cold atoms:
ICFO, MPQ, Cornell, ….
D-theories / Quantum link models in a nutshell
U(1) Quantum link model
U(1) Wilson’s theory
fermions
parallel transporters
†
Uij = e
fermions
Uij =
†
iaAij
Eij =
z
Eij = Sij
i@/@(aAij )
†
[Uij , Uij
]
Gauge fields
spanning finite-d
Hilbert spaces
+
Sij
†
[Uij , Uij
] = 2Eij
=0
[Uij , Eij ] = Uij
[Uij , Eij ] = Uij
Gauge invariance
Gj = nj +
H=
X
⇤
X
[Ej,j
~
k
[U⇤ +
†
U⇤
]
+t
Ej,j+~k ]
~
k
X
<ij>
Gauge invariance
[
†
i Uij
[Gj , H] = 0
i
+ h.c.]
Gj = nj +
H=
X
⇤
X
[Ej,j
~
k
[U⇤ +
†
U⇤
]
+t
Ej,j+~k ]
~
k
X
<ij>
[
†
i Uij
[Gj , H] = 0
i
+ h.c.]
D-theories: historical note
First proposal: Horn, 1981 (SU(2), U(1))
Proposed in condensed matter for explaining some features of
High-Tc (dimer models): Rokshar and Kivelson, 1988 (U(1))
Rediscovered independently by Orland and Rohrlich, 1990
(U(2))
Full general formulation by Brower, Chandrasekharan and
Wiese, 1995-1999 (U(N), SU(N), etc…)
Widely used nowadays in condensed matter and quantum
information: loop models, stabilizer codes, dimer models, …
Obvious key questions
Global symmetries, fermions?
As in a Wilson LGT. All fermions can be used (Kogut-Susskind,
Wilson, domain wall, …)
Which gauge
symmetries?
Lattice,
dimensionality?
U(N), SU(N),
SO(N), ZN, ….
Can be defined an
any lattice - care
with non-bipartite
(Abelian)
Continuum limit???
Care must be
taken - known for
QCD
Brower, Chandrasekharan and Wiese, Phys. Rev. D 1999. U.-J. Wiese, Annalen der Physik 525, 777 (2013).
Working example: Schwinger model
quark
anti-quark
flux string
meson
Brower, Chandrasekharan and Wiese, Phys. Rev. D 1999. U.-J. Wiese, Annalen der Physik 525, 777 (2013).
Analog quantum simulator: some setups
tB
H=
tf (c†1 c2
+ h.c.)
tB (b†1 b2
X
nF,j nB,j
nB,1
nB,2
+ h.c.) + U
U
j
U
tF , tB
tF
z
S1,2
=
Schwinger
representation:
Exactly
the building+block †we
S = b2 b1
need! 1,2
2
effective exchange Hamiltonian:
Result
population imbalance between left and right well
Hhop = J
spin imbalance between left and right well
Vshort = 6Er
t/U = 1.25
X⇣
†
†
S
x+1 x,x+1
x
Local conserved quantity!
x
+ h.c.
⌘
Vshort = 11Er
t/U = 0.26
Vshort = 17Er
t/U = 0.048
© S. Trotzky et al., Science, 319, 295 (2008)
Gx = nF,x + nB,x
Experimentally demonstrated (with bosons) (Munich, JQI)
Alkaline-earth atoms: U(N)/SU(N) setups
Color index
Fermions
Gauge fields
( 1
i = "#
brg
Local SU(N)
invariance!!
: U (1)
: U (2)
: U (3)
for SU(3) or U(3) theories
G. Pagano et al.,
Nat. Phys 2014
A Kaleidoscope: gauge theories in synthetic systems
Ultracold atoms
x
x
1
x+1
Circuit QED
U
t̃
t̃
Tewari et al. (PRL2006);
Kapit and Mueller, PRA 2010
Banerjee, MD et al. PRL 2012, PRL 2013
Stannigel et al., PRL 2014
Zohar et al, PRL 2012, PRL 2013, PRA2013
Notarnicola et al., Meurice et al., arxiv.2015;
U
Byrnes and Yamamoto, PRA 2006.
Weimer et al., Nat. Phys. 2010.
Tagliacozzo et al., Ann. Phys. 2012,
Tagliacozzo et al., Nat. Comm. 2013
A. Glätzle, MD et al., PRX
2014; arXiv2014
Trapped ions
ΩS', ωph+δph
Ωσ', ωph+δph
ΩK', ωσ+δωσ’
Marcos et al., PRL 2013,
Ann. Phys. 2014
Works in Bilbao
Rydberg atoms
Digital approaches
Hauke, MD et al.,
PRX 2013; Nath, MD
et al. arxiv.
1504.01474
t̃
+V
σ1 –V S12 +V σ2 –V
ωσ
ωS
ωσ
y
–V/2
K
x
z
Rydberg atoms,
electric dipoles
Conclusions
Dynamical gauge fields: a good playground for
exploring particle physics phenomena in AMO
Quantum links: practical framework for
implementations
U(1) theories: many platforms, building block
experiments are already along the way
U(N) / SU(N): more challenging, require ‘special’
atomic species (Florence, Kyoto, Munich,
Amsterdam, JILA,…)
And perspectives
Nuclear physics in
SO(3) models
CP(N) models
using Alkalineearth-like atoms
Open questions and challenges:
1) Wilson’s theories?
2) Plaquette terms?
3) Simpler implementations
4) Light-from-chaos and loss of gauge
invariance (for Abelian theories)
5) Gauge theories as open quantum
systems
IQOQI / ITP Univ Innsbruck
Alex Glätzle Rejish Nath
Uwe-Jens Wiese
Peter Zoller
MPK Dresden
Roderich
Moessner
Einstein Institute / ITP Univ Bern
Ioannis
Rousochatzakis
I. Bloch, C. Gross
(MPQ), R. Gerritsma, F.
Schmidt-Kaler (Mainz),
R. Blatt’s group
(Innsbruck),
K. Stannigel, D. Marcos,
P. Hauke (Innsbruck),
M. Hafezi (JQI),
S. Diehl (Dresden)
Pascal Stebler
Michael Bögli
Bilbao
Madrid
Enrique Rico
Markus Müller
Thank you
Atoms U(1): Banerjee, MD, Mueller, Rico, Stebler,
Wiese, Zoller, PRL 109, 175602;
Atoms SU(N)/U(N): Banerjee, Boegli, MD, Rico, Stebler,
Wiese, Zoller, PRL 110, 125303;
Quantum Zeno: Stannigel, Hauke, Marcos, Hafezi,
Diehl, MD, Zoller, PRL 112, 120406;
Trapped Ions: Hauke, Marcos, MD, Zoller, PRX 3,
041018 (2013);
Ions in 2D: Nath, MD et al., arXiv.1504.01474
Rydberg atoms: Glaetzle, MD et al., PRX 4, 041037;
and PRL 114, 173002
TN approaches: Rico, Pichler, MD, Zoller, Montangero,
PRL 112, 201601.
Debasish Banerjee
Review on dynamical gauge fields and
quantum simulation: U.-J. Wiese,
Annalen der Physik 525, 777 (2013).